It is time to learn what page numbers really are: fields. The gallery of page numbers in Word 2013 is nothing more than some building blocks that contain a Page field and insert it in a header or footer with additional formatting.

If it is in a footer, go to a page with the page number you want to change and under the Insert Tab, choose Footer > Edit Footer. Select the page number and eliminate the line. It is probably a paragraph border.

If instead of editing, you simply insert a different page number, that will probably get rid of the line as well, depending on which building block you choose.

Page Numbers in Word 2007-2013